16 BAIRE'S THEOREM AND ITS CONSEQUENCES 97 28. Let /be an indefinitely different iable real-valued function defined on an open interval ]a, b[ of R. Suppose that for each point x e ]a, b[ there exists an integer N(JC) such that ~D™(x)f(x) = 0. Show that /is a polynomial. (One may proceed as follows: (a) Let G be the open set of points jc e ]a, b[ such that in some neighborhood of x the function / coincides with a polynomial, and let F be the complement of G in ]a, b[. Show that F has no isolated points (3.10.10). (b) For each integer «, let EB be the closed subset of F consisting of the points x e F such that Dn/(X) = 0. If F is not empty, show that there exists a nonempty open interval I c ]a, b[ and an integer N such that F n I is nonempty and is contained in EN (use Baire's theorem). Then deduce from (a) that F n I <= En for all n > N. (c) Deduce from (b) that F n I is nowhere dense in I, and then that DN/(JC) = 0 on every component interval of G n I. Hence derive a contradiction of the hypothesis 29. Let E, F be two separable Banach spaces, F' the dual of F. Suppose that F' is contained in a Hausdorff locally convex space G, and that the topology on F' induced by that of G is coarser than the weak topology of F'. Let u : E -> G be a continuous linear mapping. (a) Show that, for every ball B in the Banach space F', the inverse image w~x(B) is a closed subset of E. (Use (12.15.9) and (12.15.8.1).) (b) Suppose that there exists a nonmeager subset A of E such that u(x) e F' for all x e A. Show that u(E) c F' and that u is continuous for the topology defined by the norm on F'. 30. Let E be a Fr6chet space, and let (wa) be a family of continuous linear mappings of E into a normed space F. Suppose that there exists a nonmeager subset A of E such that for each x e A the set of points ua(x) is bounded in F. Show that the family (w«) is equicontinuous. (For each integer n g: 1 , consider the set of points x e E such that ll««(*)ll ~ n f°r eacn index a.)ecessary and sufficient condition for the equation